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Month: February 2015

“Algorithm analysis can answer questions about running time of a standalone process, but it cannot answer questions about the performance of a system of processes competing for resources.” This is because things like storage access, i/o, and internet connections can cause delays that are difficult to predict. Queueing theory offers a way to describe (and predict) the time costs associated with such delays.

Queueing Theory Meets Computer Science

In 1909, A. K. Erlang found that the inter-arrival times for phone calls were distributed so that the probability of an inter-arrival time exceeding t decayed exponentially:

P(T>t) = exp(-λt),

with 1/λ the average time between calls. He also found that the length of phone calls were distributed the same way:

P(T>t) = exp(-μt).

Erlang employed Markov Chains (with the number of current calls as the state) to predict the probability of losing a call.

Capacity planning describes how to keep queues from getting too large, in order to prevent delays, and manage response time.

Denning gives the example of a wine cellar to demonstrate the principle of Little’s law. If a restaurant owner would like their wine to be aged R = 10 years, and they sell X = 7,300 bottles per year, the owner should build a wine seller with a Q = R * X = 73,000 bottle capacity.

Balance Equation

λ(n): Arrival rate when system is in state n.
μ(n): Completion rate when system is in state n.
p(n): Fraction of time system is in state n.

Balance Equation: For a system of states {0, 1, … n}, the balance equation is λ(n-1)p(n-1) = μ(n)p(n)

ATM:

p(n) = p(n-1)λ/μ

Probability of dropping telephone calls:

λ: Arrival Rate
1/μ: Average Call Duration
nμ: Departure Rate

p(n) = p(n-1)λ/(nμ)

Probability of dropping a call is p(N+1) where N is the capacity.

Computing with Models (Multi-Server Models)

In 1973, Jeff Buzen discovered “Mean Value Analysis”, a way to calculate in O(#users * #servers) time the server response times, system response time, system throughput, and server queue lengths. The model is extended from the principles of the operation laws.

A “hot” concept in program optimization is hotness. Cache optimization, however, has to target cold data, which are less frequently used and tend to cause cache misses whenever they are accessed. Hot data, in contrast, as they are small and frequently used, tend to stay in cache. The “coldness” metric in this paper shows how the coldness varies across programs and how much colder the data we have to optimize as the cache size on modern machines increases.

For a program p and cache size c, the coldness is the minimal number of distinct data addresses for which complete caching can obtain a target relative reduction in miss ratio. It means that program optimization has to improve the locality for at least this many data blocks to reduce the miss ratio by r in size-c cache.

coldness(c, r) = (−1) ∗ (#uniq addr)

The coldness shows that if the program optimization targets a small number of memory addresses, it may only be effective for small cache sizes and cannot be as effective for large cache sizes.

For 10% miss reduction, the coldness drops from -15 for 1KB cache to -4630 for 4MB cache in integer applications. In floating point applications, it drops from -4 for 1KB cache to -63,229 for 4MB cache. Similarly, for 90% miss reduction, the coldness drops from -11,509 to -50,476 in integer applications and from -562,747 to -718,639 in floating point applications. For the 4MB cache, we must optimize the access to at least 344KB, 2.4MB, and 5.4MB data to reduce the miss ratio by 10%, 50%, and 90% respectively. In the last case, the coldness shows that it is necessary to optimize for a data size more than the cache size to obtain the needed reduction.

Next is the experimental data that produced these coldness data. It shows the minimal number of distinct addresses that account for a given percentage of cache misses.

The average number of most missed addresses increase by about 100x for top 10% and 50% misses as the cache size increases.

Based on the individual results, we classify applications into two groups. Applications that are consistently colder than the me- dian are known as below median cold and applications that are consistently not as cold as the median are known as above median cold applications.

Two Related Metrics

Program and machine balance. The limitation can be quantified by comparing program and machine balances [Callahan+:JPDC88]. For a set of scientific programs on an SGI Origin machine, a study in 2000 found that the program balances, ranging from 2.7 to 8.4 byte-per- flop (except 0.04 for optimized matrix multiplication), were 3.4 to 10.5 times higher than the machine balance, 0.8 byte-per-flop. The maximal CPU utilization was as low as 9.5%, and a program spent over 90% of time waiting for memory [DingK:JPDC04].

A rule of thumb is that you halve the miss rate by quadrupling the cache size. The estimate is optimistic considering the simulation data compiled by Cantin and Hill for SPEC 2000 programs, whose miss rate was reduced from 3.9% to 2.6% when quadrupling the cache size from 256KB to 1MB.

“Tuesday morning lows were running 30 to 40 degrees below average from Indiana to New England. Dozens of daily record lows fell Tuesday morning, by as much as 20 degrees. … A few readings have broken century-old records, including those in Pittsburgh; Akron-Canton, Ohio; Hartford, Conn.; and Indianapolis. In Rochester, N.Y., the low of minus-9 degrees tied the record set in 1889. Records in Rochester go back to 1871. … The western suburbs of Washington had their coldest morning in nearly two decades. Dulles International Airport set a record low of minus-4 degrees, which broke the previous record set in 1967 by 18 degrees.”

While programs running together, fairness become an important metric beyond throughput. In this paper, authors propose 1) a definition for fairness and 5 possible metrics.2) 2 algorithms to improve the metrics through cache partition.

According to simulation experiments, authors show that 1) 2 of 5 metrics are not strong correlated to their definition of fairness, 2)LRU or Pseudo-LRU is not fair, 3) Improving fairness usually equal to improving throughput and 4)their cache partition algorithms improve fairness by a factor of 4X while increasing throughput by 15% on average.

More specifically, authors define ideal fairness to be

Where denotes execution time of program i while co-running and for solo-run. Then authors quantitive fairness to be

However, execution time is difficult to measure because of the lack of reference points in the execution. Thus 5 possible metrics are proposed in section 2.3. According to the correlation between these 5 metrics and fairness, turns out 2 metrics are not strong related to fairness and others are sufficient enough. At last authors selected the most correlated metric as following:

After metric is decided, authors proposed 2 algorithms to optimize fairness statically or dynamically. Both of them need additional hardware counter or functionalities. In brief, the static algorithm partition cache before programs running according to stack analysis results, which is gained through profiling. And the dynamic algorithm sampling programs periodically and adjust partition according to measures of sampling phase.

In evaluation, 18 2-programs pairs are tested, the 13 programs come from SPEC2K. A simulator is used with additional hardware counter and functionalities implemented.

Metric correlation is evaluated firstly. They figure out average correlation for all metrics and based on that, they select the best metric which has 94% correlation with fairness.

Then according to comparing between LRU and static partition, LRU produce very unfair sharing in 15 of 18 pairs, and in 3 pairs, LRU is better on fairness. Only in 1 pair, LRU gains better throughput.

The results for dynamic partition is similar. Even though Pseudo-LRU is slightly better than LRU on both throughput and fairness, but it still worse than dynamic partition with only 1 exception, in which PLRU is almost ideal.

The StatCache which proposed by Berg and Hagersten is a low overhead statistical model. It can estimate miss ratio for caches of arbitrary size. But this model only report the average miss ratio which may not fully expose the application’s features. The miss ratio R can be calculated by the following function:

R * N = h1 f(1*R) + h2 f(2*R) + h3 f(3*R) + …

N = h1+ h2 + h3 + …

f(n) = 1 – (1-1/L)n

N is the number of reuse distance samples, f(n) is the function that indicate the probability that a cache line has been evicted from the cache when it was in the cache n cache misses ago. L is the number of cache lines in the cache.

One simple solution is that divide the program into window and sampling for each window to handle program phases. But the overhead will be significantly increase to capture the short application phases. To avoid this overhead, phase detecting algorithms are used to divide the program into different phases, sampling each phase once is enough and the subsequent instances of the same phase can use the same profiled data.

Based on this thinking, They proposed the ScarPhase (Sample-based Classification and analysis for Runtime Phases). They group the reuse distance of the program by phases and than apply the StatCache model to calculate the miss ratio for each group. They use ScarPhase library to detect and classify phases base on the application’s execution history.

Also they found that variations can also happen inside phases and classified them into four types: No variations, Transition variations, Periodic variations and Instance variations.

(1)No variations mean little changes happened inside the phase, so only a small part of each phase needs to be profiled.

(2)Transition variations mean the behavior slowly changes form one phase to another. So the transition should be divided into windows from the start to the end. More windows to profile, more accuracy we gain.

In this chapter of the book, some techniques in optimization of loop-based programs’ prallelism and locality are discussed. It starts with an introduction of the basic concepts: the affine access pattern of the data references. Three types of spaces: the iteration space, the data space and the processor space. The essential problem of optimization is to increasing loop’s parallelism and locality, by building multidimensional spaces and affine mappings between these spaces and transforming the loops accordingly.

The chapter is divided into two parts: first, introducing the model of the loops and second, introducing the techniques of

optimization.

Start from the first part. First, how the loop is structured. The loop nests are structured as a multidimensional vector space. The loop bounnds and accesses are represented as set of affine constraints, which can be abstracted as a matrix-vector multiplication form. Symbolic constants (or parameters) are introduced and can be modeled. Next, only the affine accesses can be modeled and they represent many programs in real world. Third, data reuse is categorized into two types: self reuses and group reuses. Self reuse means the reuse is from the same ‘static’ access, while group reuse means the reuse is from different ‘static’ access. The self reuses bring huge savings on memory accesses and the amount of savings can be calculated as the rank of the coefficient matrix in the access. Group reuses are actually a more interesting case, but its analysis limited to accesses whose coefficient matrix are the same. The fourth, representing data dependence. Three types of dependencs are first introduced: true dependence, anti dependence, and output dependence. Integer linear programming is a general solution of finding dependences, but it is an NP-complete problem. There are three parts of solving the integer linear programming problem. First, check the existence of the solution, by using Greatest Common Divisor (GCD) test. Second, try a set of simple heuristics to handle the large amount of inequalities. If the heuristics fail, try integer programming solvers using branch-and-bound.

The second part of the chapter is optimization based on the first part. Three sections are used to cover parallelization for minimizing synchronization. Three steps are required to perform the parallelization: first, split the computations into many independent units; second, assign the computation units to the processors; third, generate an SPMD program that executes on multiple processors. The first problem that is looked at in the book is the problem of finding parallelism that requires no synchronization. The solution is to respect the constraints of data dependence by assigning the operations that share data on one processor. In practice, software pipelining is used to partition programs that minimizes the synchronization between processors. (need more details on this part)

The last part of optimization is to optimize for locality. Three techniques are introduced in the book for both uniprocessors and multiprocessors. First exploiting temporal locality. It basically schedules computations that share data close enough in execution. This part in the book is rather general, only an example is provided to demonstrate the concept. Second, array contraction. When a sequential execution that operates on one array element at a time serially, the array can be contracted by replacing the intermediate array with a scalar. It reduces the need for data storage, but also reduces the parallelism available. The third, interleave the partitions. Two primitives can be employed to reduce the distances between reuses across different iterations. These two primitives can be repeated. Interleaving inner loops in a parallel loop, known as ‘strip-mining’. It blocks the outer loop and moves the blocked loop inside the inner loop to create more reuses within the blocks. Interleaving statements in a parallel loop. This transformation distributes a ‘stripmined’ loop across the statements. It shortens the distance between one statement within a loop using blocking. It is effective when there is a large loop ‘interferes’ the data reuse of one statement in the loop.

There are other forms of affine transformation, mainly targeting on distributed memory machines, multi-instruction-issue processors and vector/SIMD instructions. Prefetching is also briefly mentioned.

While improving locality through cache management, one of the most significant problems is determining how close is our approach to the optimal one. For sequential program, we get OPT replacement algorithm. And for co-runed programs, we should have one. This paper tried. Authors figure out a model on analyzing quality of partition, figuring out optimal partition and dynamically close to it.

This paper contributes in 4 aspects:

Optimally partition cache for 2 programs

Find a partition which has similar performance as LRU

Theoretically show LRU is far from optimal for transient data allocation

Near optimally partition cache for N programs

Optimally partition cache for 2 programs

Authors start modeling with an insight that miss ratio is linear to cache size in log/log scaling. Thus they got a prediction function for miss ratio that MI(x)=aI*x^(bI*log10), where x is cache size assigned to program I, a and b are coefficient which got from profiling.

Since the total misses is TotalMisses=(MI(x)+MD(C-x))*T/2, where C is total size of cache, and they suppose the two programs I and D has same access rates T. Then within an assumption that the miss ratio is convexity according to cache size, they can got minimal point for TotalMisses when derivation equal to 0.

To achieve this goal, authors firstly propose a concept named state of cache for 2 programs. A State x of cache means overall miss ratio without partition is equivalent to a partition, whichassign x cache for program I and C-x for program D. And S(x) is possibility function of state x appears.

Since the possibility of stat x transforms tox+1 should be equal to the possibility of x+1 to x, then they have such equation that S(x)MI(x)=S(x+1)MD(C-x-1). According to the monotonic property of MI and MD, S(x) is unimodal.

Finally, for the most possible stat x’, authors showed it is close to optimal partition.

LRU is far from optimal for transient data allocation

With derivation of stats according to time, authors shows LRU needs time to achieve state x’, which is the most possible state as well as the state near to optimal partition.

Since derivation of state x can be represent as dx/dt = Rate of increasing x – Rate of decreasing x,which is essentially equal to MI(x) – MD(C-x). This derivation shows velocity and time for LRU to achieve a near optimal partition state, and also shows when time is short enough, LRU cannot perform close to optimal.

Near optimally partition cache for N programs

With the assumption that miss ratio function is convexity, authors propose a greedy algorithm to achieve a near optimal solution according to the insight of the first section, which is optimal is achieved when the sum of derivation of miss ratio equal to 0. Thus they employ followed algorithm keep the sum as close to 0 as possible:

Let Ci to be cache size of partition i

Initialization: set C1=C2=…=CN=0

Induction step : Find the most benefit programs i, which would reduce the most cache misses with extra 1 cache block

let Ci=Ci+1, and keep other partition does not change, goto step 2 until sum of partition size equal to C

The reason is thatthe performance of a parallel program is dependent on many factors. Mark Edward Crovella who was awarded Phd degree by URCS in 1994 divided the factors into two groups: external factors and internal factors.

External factors are the aspects of the program’s run-time environment which is not specified in the parallel program:

1: the number of processors used to run the program

2: the size of the input data set

3: the structure of the input data set (sparseness of a matrix)

4: the problem definition (a program can solve multiple variations of a problem)

5: the machine used to run the program (architecture)

Internal factors are the methods used to parallelize the application. Typical examples of internal factors are:

1: type of parallelism used

2: which code to parallelize

3: synchronization methods

The external factors will determine the choice of internal factors. Programmers should not expect to find one general best structure for parallel program. With these insights, he proposed three techniques to assist rapid development of parallel efficient programs: Predicate profiling, lost cycles analysis and lost cycles modeling.

Predicate profiling:

The standard formula to measure the performance of parallel program is described below. But To(n,p) is not specified which is not so helpful for optimization. Because the reasons cause the overhead are not exposed.So he decomposed To(n,p) into performance predicates.

In there states, no useful work is down, so they call the time spent in the stats Lost Cycles (LC). So he model each component of To instead of modeling To which will reduce the difficulty in modeling.

To(E) = IP(E) + LI(E) + SL(E) + CL(E) + RC(E)

IP, LI, SL, CL, RC are functions of n and p with parameters. And the parameters are determined by the predicate profiling data. With parameters set down, the components of theoverhead can be exposed which can guide the optimization.